Question

in a triangle pqr median ps is produced to a point t such that ps=st. prove that pqtr is a parallelogram​

Answer 1

PQTR is a parallelogram

Step-by-step explanation:

PS is median

=> QS = SR

PS = ST given

Comparing Δ PSR & Δ TSQ

PS = TS

SR = SQ

∠PSR = ∠TSQ ( vertically opposite angles)

=> Δ PSR ≅ Δ TSQ

=> QT = PR

& ∠QTS = ∠RPS

=> ∠QTP = ∠RPT

=> QT ║ PR

Similarly

ΔPSQ ≅ ΔTSR

=> PQ = RT

& ∠QPS = ∠RTS

=> ∠QPT = ∠RTP

=> PQ ║ RT

QT ║ PR & PQ ║ RT

QT = PR & PQ = RT

=> PQTR is a parallelogram

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Answer 2

Step-by-step explanation:

In ∆PSQ and ∆TSR we get

=> ST = PS ( Given )

=> QS = SR ( median PS divides QR equally )

=> <PSQ = <TSR ( vertically opposite angle )

so , by SAS congruence rule ∆PSQ is congruent to ∆TSR

NOW , <QPS = <STR by CPCT

but this is also called the alternative angle of two line QP & TR so according to that as the alternative angle of two lines are equal ultimately QP//TR .......... hence proved

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